A weighted Sobolev space theory of parabolic stochastic PDEs on non-smooth domains

Abstract

In this paper we study parabolic stochastic partial differential equations defined on arbitrary bounded domain $\cO \subset \bR^d$ allowing Hardy inequality: $$ \int_{\cO}|\rho^{-1}g|^2\,dx\leq C\int_{\cO}|g_x|^2 dx, \quad \forall g\in C^{\infty}_0(\cO), $$ where $\rho(x)=\text{dist}(x,\partial \cO)$. Existence and uniqueness results are given in weighted Sobolev spaces $\frH^{\gamma}_{p,\theta}(\cO,T)$, where $p\in [2,\infty)$, $\gamma\in \bR$ is the number of derivatives of solutions and $\theta$ controls the boundary behavior of solutions. Furthermore several H\"older estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.